Articles of trigonometry

solve a trigonometric equation $\sqrt{3} \sin(x)-\cos(x)=\sqrt{2}$

$$\sqrt{3}\sin{x} – \cos{x} = \sqrt{2} $$ I think to do : $$\frac{(\sqrt{3}\sin{x} – \cos{x} = \sqrt{2})}{\sqrt{2}}$$ but i dont get anything. Or to divied by $\sqrt{3}$ : $$\frac{(\sqrt{3}\sin{x} – \cos{x} = \sqrt{2})}{\sqrt{3}}$$

Computation of $\int _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$

I need to compute $$\int \limits _{-\pi} ^\pi \frac {e^{in\theta} – e^{i(n-1)\theta}} {\mid \sin {\theta} \mid} d\theta .$$ Does anyone see any good strategy? Thanks.

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$

If $\sin x + \sin^2 x =1$ then find the value of $\cos^8 x + 2\cos^6 x + \cos^4 x$ My Attempt: $$\sin x + \sin^2 x=1$$ $$\sin x = 1-\sin^2 x$$ $$\sin x = \cos^2 x$$ Now, $$\cos^8 x + 2\cos^6 x + \cos^4 x$$ $$=\sin^4 x + 2\sin^3 x +\sin^2 x$$ $$=\sin^4 x […]

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ I will be satisfied with an answer involving special functions (e.g. modified Bessel function). I tried expanding the product terms using the product-to-sum identities, as well applied other trigonometric identities, to no avail. Can anyone help? I suspect that the solution will involve […]

Calculate the angle of a vector in compass (360) direction

Given a vector described by X and Y components $x=4$ $ y=-2$ I get the direction of the vector by using $\arctan{-\frac 24}$. $\theta= -0.4636 \text{ rad}$ $\theta= -26.565^{\circ}$ My initial confusion was that I assumed the degrees in theta referred to a compass direction as would be measured with a protractor. But a vector […]

Solve $\tan\left(\arccos\left(\frac{-\sqrt{2}}{2}\right)\right)$ without calculator

This is a question from the practice exercises of Barron’s AP Calculus. The directions state that I cannot use a calculator. (No trig tables I think, since those are not allowed in the exam) So, how to solve $$\tan\left(\arccos\left(\frac{-\sqrt{2}}{2}\right)\right)$$ without using a calculator? (With a knowledge of basic trig values like 0,30,45,60,90)

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y – \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks…

Cubic trig equation

I’m trying to solve the following trig equation: $\cos^3(x)-\sin^3(x)=1$ I set up the substitutions $a=\cos(x)$ and $b=\sin(x)$ and, playing with trig identities, got as far as $a^3+a^2b-b-1=0$, but not sure how to continue. Is there a way to factor this so I can use the zero product property to solve? Thanks for any help/guidance! P

Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$

For any $z=x+iy$, prove the following: $$|\sin z| \geq |\sin x|$$ $$|\cos z| \geq |\cos x|$$ $\epsilon$-$\delta $ proof is not required. I don’t really know how to proceed. I know in order to remove the absolute values I can square both sides and I have tried proving this statement using the hyperbolic forms and […]

$x^{2}+2x \cdot \cos b \cdot \cos c+\cos^2{b}+\cos^2{c}-1=0$

Solve this equation : $$x^{2}+2x \cdot \cos b \cdot \cos c+\cos^2{b}+\cos^2{c}-1=0$$ Such that $a+b+c=\pi$ I don’t have any idea. I can’t try anything.